3c.What is
a unitarity triangle?

guided tour

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(If you are not familiar with imaginary and complex numbers, skip this page)
The CKM matrix has a property known as unitarity due to the physical properties it represents. Mathematically, this defines a set of equations that the matrix elements must satisfy. For one such equation, take each element of the first column of the matrix, multiply it by the complex conjugate of its neighbor in the third column, and then sum the three products and set it equal to zero: V_udV_ub^* +V_cdV_cb^* +V_tdV_tb^* = 0 We obtain an equivalent equation if we divide both sides of the equation by the middle term:

V_udV_ub^*V_cdV_cb^*

+ 1 +

V_tdV_tb^*V_cdV_cb^*

= 0

Complex numbers can be represented as points on a two-dimensional plane, or equivalently as vectors in a plane. Each term in the above equation is complex and can be drawn as a vector in the plane. Arranging the three vectors head-to-tail gives the sum. Since the sum is zero, the three vectors should form a closed polygon, that is, a triangle. Since one leg of the triangle is just 1.0, it lies on the x-axis and has a length of 1. The triangle is then defined by a single additional point on the plane, which we specify by the coordinates (r,h). This is known as a unitarity triangle. The three angles f₁, f₂, and f₃ are also known as b, a, and g, respectively.